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Full text of "Analytical Mechanics"

to the wall.   Therefore denoting the lengths of the beam and the string by I and a, respectively, we have
2X ** R* - T = 0,                                T
SF s & - W= 0,
where SGV denotes the sum of the moments of the forces about an axis through the point 0' perpendicular to the x?/-plane. Solving the last three equations we have
T =
2  A/22-a2
FIG. 33.
DISCUSSION.  It should be noticed that in taking the moments the axis was chosen through the point Or in order to eliminate the momenta of as many forces as possible and thus to obtain a simple equation.
The reaction Ri is independent of the angular position of the beam and equals the weight W. On the other hand R2 and T vary with a.
When a = - both R2 and T vanish.   As a is diminished from ~ to 0, Rs
J                                                                                                                                          2
and T increase indefinitely.
2. A ladder rests on rough horizontal ground and against a rough vertical wall. The coefficient of friction between the ladder and the ground is the same as that between the ladder arid the wall. Find the smallest angle the ladder can make with the horizon without slipping.
There arc three forces acting on the ladder, i.e., its own weight W and the two reactions Ri and R2. Replacing Ri and R2 by their components and writing the equations of equilibrium we obtain
il cos a + N%1 sin a  W ~ cos a = 0,
where a is the required angle. We have further